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Theory of Networked Computation?

Joan Feigenbaumhttp://www.cs.yale.edu/homes/jf

DELIS Workshop; Barcelona; February 2008

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Future of Distributed Computing?

Conjecture: “Distributed-computing research” will be more like “networking research.”

What characterizes “networking research”?

• “Sacrifice strict semantics for scalability.”[Scott Shenker, PODC 2003]

• “Evolutionary fitness trumps elegance.”[Jonathan Smith, Colloquium talk 2007]

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Why is this Plausible?

• It’s already happening; see PODC and EC proceedings of the last five years.

• Cultural trend: “networkization” of CS• Funding trend: various “Clean-Slate Design”

initiatives, many 100s M$ worldwide• Networks provide real-world examples of

distributed computations.• Intellectually compelling

Role of the theory community?

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Elements of a “Theory of NC”

• Model(s) of computation• General network-algorithmic techniques• Algorithms for concrete problems of

interest• Lower-bound techniques, reductions• Hardness results for concrete problems

of interest Descriptive results and interpretation

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Examples of “Networked Comp.”

• Routing, congestion control, and other “network-layer” computations

• WWW search• Auctions• P2P file sharing• Blogs, wikis, MySpace, and other

“web-mediated communities”• Yahoo! questions, ESP, del.icio.us, and

other “human-aided computations”

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Properties to Model

• Massive scale• User self-interest• Subnetwork autonomy• Emergent behavior• Extreme heterogeneity (of devices, uses,

subnetworks, …) Results without convergence Agents, data, resources, and network

conditions that change during computation

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Outline

• Theory of incentive-compatible IDR

• What IDR has contributed to ToNC• What IDR has not (yet?) contributed

to ToNC• More ToNC-agenda items

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Interdomain Routing

Establish routes between autonomous systems (ASes).

Currently done with the Border Gateway Protocol (BGP).

AT&T

Qwest

Comcast

Verizon

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Why is Interdomain Routing Hard?

• Route choices are based on local policies.

• Autonomy: Policies are uncoordinated.

• Expressiveness: Policies are complex.

AT&T

Qwest

Comcast

Verizon

My link to UUNET is forbackup purposes only.

Load-balance myoutgoing traffic.

Always chooseshortest paths.

Avoid routes through AT&T ifat all possible.

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BGP Route Processing (1)

• The computation of a single node repeats the following:

Receive routes from neighbors

UpdateRoutin

g Table

Choose“Best”Route

Send updatesto neighbors

• Paths go through neighbors’ choices, which enforces consistency.

• Decisions are made locally, which preserves autonomy.

• Uncoordinated policies can induce protocol oscillations. (Much recent work addresses BGP convergence.)

• Recently, private information, optimization, and incentive-compatibility have also been studied.

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Apply Policy =filter routes & tweak attributes

BGP Route Processing (2)

Routing Table

Apply Import Policies

Best Route Selection

Apply Export Policies

Install forwardingentries for best routes

ReceiveBGPupdates

Storageof routes

TransmitBGP updates

Based onattributevalues

IP Forwarding Table

Apply Policy =filter routes & tweak attributes

Open-ended programming:constrained only by vendor configuration language

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Example: Convergence

1 2

d

Prefer routes

through 2

Prefer direct route

to d

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Example: Oscillation

1 2

d

BGP might oscillateforever between

1d, 2dand

12d, 21d

Prefer routes

through 2

Prefer routes

through 1

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Example: Convergence

1 2

d

Prefer routes

through 2

Prefer routes

through 1

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Dispute Wheels

Nodes ui, hub routes Ri, and spoke routes Qi.

Each ui prefers RiQi+1 to Qi.

“No dispute wheel” —>robust convergence

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Gao-Rexford Framework (1)

Neighboring pairs of ASes have one of:– a customer-provider relationship

(One node is purchasing connectivity fromthe other node.)

– a peering relationship(Nodes have offered to carry each other’stransit traffic, often to shortcut a longer route.)

peer

providers

customerspeer

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Gao-Rexford Framework (2)

• Global constraint: no customer-provider cycles• Local preference and scoping constraints, which are

consistent with Internet economics:

• Gao-Rexford conditions => BGP always converges [GR01]

Preference Constraints

. . . . . .

. . . . . .i

dR1

R2

k2

k1

• If k1 and k2 are both customers, peers, or providers of i, then either ik1R1 or ik2R2 can be more valued at i.

• If one is a customer, prefer the route through it. If not, prefer the peer route.

Scoping Constraints

d

k

i

j

• Export customer routes to all neighbors and export all routes to customers.

• Export peer and provider routes to all customers only.

m

. . . .. . . .

. . . .peer

customer

provider

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Open Problem 1: Convergence

Fully characterize the conditions under which BGP converges (robustly).

• “No dispute wheel” is sufficient but not necessary. Is it enforceable?

• Develop game-theoretic and algorithmic frameworks that explain how dispute wheels form and how to avoid them. (See Levin-Schapira-Zohar in STOC ’08.)

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Open Problem 2: Round Complexity

On those instances on which BGP is guaranteed to converge, how many rounds does it take to converge?

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Economic Mechanism Design

Agents: 1, …, nStrategies: s1, …, sn

Types: t1, …, tn

Actions: a1, …, an

Outcome: oValuation functions: v1, …, vn

Payment functions: p1, …, pn

Utility functions: u1, …, un

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Classical, One-Round Mechanisms (1)

o

Agent 1

a1 s1(t1)

Agent n

an sn(tn) • • •

a1 p1 • • • an pn

t1 tn

Mechanism

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Classical, One-Round Mechanisms (2)

• Action vector (a1, …, an) is “consistent with selfishness.”– ai maximizes ui(o, ti) vi(o, ti) + pi.

– Meaning of “maximize” depends on “solution concept,” e.g., NE, BNE, DSE, epNE, …

• Mechanism-design goal: o(a1, …, an) G(t1, …, tn)

• Classical economic-MD question: For a given solution concept, which design goals can be achieved?

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Solution Concepts (1)• (a1, …, an) is a Nash Equilibrium (NE) if

i and ai : ui(o(a1, …, ai, …, an), ti)

ui(o(a1, …, ai, …, an), ti)– Given other agents’ actions, agent i is best off playing

ai.

• (a1, …, an) is a Dominant-Strategy Equil. (DSE) if i, ai, and (a1, …, ai-1, ai+1, …, an):

ui(o(a1, …, ai, …, an), ti) ui(o(a1, …, ai, …, an), ti)– Regardless of other agents’ actions, agent i is best off

playing ai.

‘‘‘,

,

,

,

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• (s1, …, sn) is an ex-post Nash Equil. (epNE) if

i, si, and (t1, …, tn):

ui(o(s1(t1), …, si(ti), …, sn(tn)), ti)

ui(o(s1(t1), …, si (ti), …, sn(tn)), ti)

– Given that other agents follow the prescribed strategy, agent i is best off doing so, too, regardless of the other players’ types.

Solution Concepts (2)

,

,

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Algorithmic Mechanism Design [NR01]

• Required polynomial-time o( ) and p( ).

• Focused on strategyproof, direct-revelation mechanisms.

• Put forth polynomial-time, strategyproof LCP-routing MD as a good abstraction of Internet routing.

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P-Time VCG Mech. for LCP [NR01]

• •x y

ei

ej

Agent i edge ei

Type ti cost(ei)

Outcome o LCP from x to y (wrt reported costs)

0 if ei o

cost(o(Gi <- )) - cost(o(Gi <- o)) if ei o{pi(o)

G

vi(o) { 0 if ei o

-cost (ei) if ei o

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Moving Closer to Real IDR [FPSS02, SP04]

• Nodes (ASes), not edges, are the agents.

• All-source, LCP tree Td to each destination d.

• No trusted center; ASes compute the routes themselves.

• Use BGP as an algorithmic substrate to preserve “evolutionary fitness” and encourage adoption.

BGP-compatible VCG mechanism for LCP trees that is incentive-compatible in

epNE

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“Optimality” wrt LCP is too narrow.

• More generally, seek Td = {R1, …, Rn} that maximizes Σi vi(Ri).

• Fully general vi (routing policies)• Next-hop policies: For all i and j,

vi(ijk3…kmd) = vi(ijl3 …ltd)• Forbidden-set policies: For every

source i, there is a set Si of transit nodes such that vi(R1) > vi(R2) if R2 contains a node in Si but R1 doesn’t.

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Policy Consistency

. . .

. . . .

d

k i

IFvk(R1) ≥ vk(R2)

R2

R1

THENvi((i,k)R1) ≥ vi((i,k)R2)

Valuations are policy consistentiff, for all routes R1 and R2

(whose extensions arenot rejected),

(analogous toisotonicity [Sobrino])

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Requirements

Results References

Lowest-cost routes BGP-compatible VCG computation

FPSS02, SP04

Fully general routing policies

NP-hard even to approximate

FSS04

Next-hop routing policies

Centralized, ptime VCG computation. Not BGP compatible

FSS04

Forbidden-set routing policies

NP-hard even to approximate

FKMS05

Gao-Rexford + policy consistency

BGP is inc.-comp. in collusion-proof epNE w/o payments

FRS06, FSS07

Gao-Rexford + route verification

BGP is inc.-comp. in collusion-proof epNE without payments.

LSZ08

Linear utilities Uncapacitated links

BGP-compatible VCG computation

HNP07

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Open Problem 3: Equilibrium

Fully characterize the conditions under which epNE is reached simply by executing BGP.

• No dispute wheel plus policy consistency are sufficient but not known to be necessary.

• No complete characterization is known even if we ignore collusion and/or allow payments.

• Related to security problems in [LSZ08]

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Open Problem 4: Better Computational Models

• “Results without convergence” (Kearns 2006)

• No fixed instance: Policies, AS graph, and network conditions change during the computation.

• Develop measures of “network complexity” that deal successfully with these and other important features of networked computation.

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Open Problem 5: Better Economic Models

• Is epNE (or even “equilibrium”) the right “solution concept” for IDR?

• The role of money- Do without payments altogether?- Incorporate payment protocols?

• Mechanisms with partially known, dynamically changing strategy spaces

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Outline

• Theory of incentive-compatible IDR• What IDR has contributed to

ToNC• What IDR has not (yet?)

contributed to ToNC• More ToNC-agenda items

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Contributions

• Pushes the envelope on incentives in computation.

• Combines the relevant research areas (algorithms, networking, and economics) in a serious way.

• Helps explains why interdomain routing “works,” despite the “proofs” that BGP is “wrong.”

• Exemplifies “protocol-based algorithms design.”

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Unresolved Issues

• Computational modeling challenges, e.g.,- Results without convergence

- AS graphs, policies, and loads that change during the computation

• Is epNE (or even “equilibrium”) really a useful concept in networked computation?

• Are there any general techniques or insights here, or is IDR unique?

• Interaction with AS-graph formation

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Talk Outline

• Theory of incentive-compatible IDR• What IDR has contributed to ToNC• What IDR has not (yet?) contributed

to ToNC• More ToNC-agenda items

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Open Problem 6: Integrating Diverse

Theories• We’ve seen the influence of TCS, Game

Theory, Distributed Computing, Networking, Cryptography, Model Checking, …

• If we try to combine all of the formalism and assumptions of these diverse fields, we will NOT be able to prove (or even state) meaningful theorems.

For each networked-computational problem, figure out what you need to be precise about and what you can fudge or ignore.

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Open Problem 7: Go Beyond TCS-Style Theorems

• We will NEVER have a fixed IDR “instance.” So what do “BGP-convergence” theorems mean?

• NC problems that are provably hard for networking reasons (i.e., not because they’re NP-hard) are few and far between.

Develop a complexity theory of networked computation: Relevant computational resources, “results without convergence,” general algorithmic techniques, canonical hard problems and reductions, …

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Open Problem 8: How General is Protocol-based

Alg. Design?• “BGP-compatible” algorithmic mechanisms

can leverage the evolutionary fitness of today’s IDR framework and would be easier to deploy than algorithmic mechanisms designed from scratch.

• Are there other pieces of the computational infrastructure that can be used in this way? Candidates: Search, keyword auctions, …

Consider the use of widely deployed, successful protocols as “computational substrates” for novel network algorithms.

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Conclusions

• Good opportunity to do novel theoretical work that has practical impact

? IDR is fascinating, but unfortunately it may be unique.

• Multidisciplinarity is exciting but creates technical challenges.

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References

• “Distributed Algorithmic Mechanism Design,’’ by F., Schapira, and Shenker [A chapter in Algorithmic Game Theory, Nisan, Roughgarden, Tardos, Vazirani (eds.), Cambridge University Press, 2007]

• “Theory of Networked Computing” webpage: http://www.cs.yale.edu/homes/jf/ToNC.html